f-t-0-t-x-x-dx-

Question Number 180786 by Vynho last updated on 17/Nov/22

f (t) = \int_{0}^{t} x - ⌊ x ⌋ d x

Answered by mr W last updated on 17/Nov/22

l e t n = ⌊ t ⌋

f (t) = \underset{k = 0}{\sum^{n - 1}} \int_{k}^{k + 1} (x - ⌊ x ⌋) d x + \int_{n}^{t} (x - ⌊ x ⌋) d x

f (t) = \underset{k = 0}{\sum^{n - 1}} \int_{0}^{1} (k + ξ - k) d (k + ξ) + \int_{0}^{t - n} (n + ξ - n) d (n + ξ)

f (t) = \underset{k = 0}{\sum^{n - 1}} \int_{0}^{1} ξ d ξ + \int_{0}^{t - n} ξ d ξ

f (t) = \underset{k = 0}{\sum^{n - 1}} (\frac{1}{2}) + \frac{{(t - n)}^{2}}{2}

f (t) = \frac{n}{2} + \frac{{(t - n)}^{2}}{2}

f (t) = \frac{⌊ t ⌋ + {(t - ⌊ t ⌋)}^{2}}{2}

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